Retopology - move vertex + snap to model
I suppose you don't see your vertex sticking to the surface because of the modifier setting: you should activate Display Modifier in Viewport and Display Modifier in Edit Mode so that you can see the original mesh and its modified version, and put a bit of Offset if you want to see it out of its Target surface.



Also, I'm not sure about the Snap option, as you already use the Shrinkwrap modifier. I guess you should either use the Snap or the Shrinkwrap method. If you use the Snap option, select Face as Snap Element and activate the other Snap options.